The reciprocal of the value of the product  $\sin \left(\frac{5\pi }{2^{11}}\right)\cos \left(\frac{5\pi }{2^{11}}\right)\cos \left(\frac{5\pi }{2^{10}}\right)\cos \left(\frac{5\pi }{2^9}\right)\cdot \cdot \cdot \cdot \cdot \cdot \cos \left(\frac{5\pi }{2^3}\right)\cos \left(\frac{5\pi }{2^2}\right)$  equals ___

We have 
 $\frac{2^{10}\sin \left(\frac{5\pi }{2^{11}}\right)\cos \left(\frac{5\pi }{2^{11}}\right)\cos \left(\frac{5\pi }{2^{10}}\right)\cos \left(\frac{5\pi }{2^9}\right)\mathrm{.....}\cos \left(\frac{5\pi }{2^3}\right)\cos \left(\frac{5\pi }{2^2}\right)}{2^{10}}$

$=\frac{\sin \frac{5\pi }{2}}{2^{10}}=\frac{\sin {450}^0}{2^{10}}=\frac{1}{2^{10}}=\frac{1}{1024}$
$\Rightarrow$Reciprocal of the value of the product =1024